310 

P4 
1914 


UC-NRLF 


9-A  SHORT  TABLE  OF  INTEGRALS 


COMPILED  BY 

B.  O.  PEIRCE 

// 

LATB  HOLLIS  PROFESSOR  OF  MATHEMATICS  AND  NATURAL  PHILOSOPHY 
IN  HARVARD  UNIVERSITY 


^^ 


ABRIDGED  EDITION 


GINN  AND  COMPANY 

BOSTON  .  NEW  YORK  •  CHICAGO  •  LONDON 
ATLANTA    DALLAS  •  COLUMBUS  •  SAN  FRANCISCO 


/D 


>     •  •  • 

•  •   •  • 

•  •    •  •• 


• .  •  •  •  • 


COPYRIGHT,  1914,  BY 
GINN  AND  COMPANY 


ALL  RIGHTS  RESERVED 

PRINTED  IN  THE  UNITED  STATES  OF  AMERICA 

329.4 


GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


m 


FUNDAMENTAL  EQUATIONS 

1.  I  a  'f(x)dx  =  a  I  f(x)dx;    I  <f>('i/)dx=  I  ^^^ dy,  where  y'=di//dx. 

2 .  I  (u-\-v)dx=j  udx-{-  j  V  dx,  where  u  and  v  are  any  functions  of  ar. 

3.  I  tcdv  =  uv  —  I  vdu  ;     lu-T-dx  =  uv—  iv-^-  dx. 
J  J  J      dx  J      dx 

4.  /  x'^dx  =  — ——7  f  if  m  ^  —  I'j     j  —  =  log x,  or  log (—  x). 

5.  I  e'^dx  =  e'^/a  ;     /  b'^dx  =  — • 


6.    I  sinxdx  =— Gosx;     lGOSxdx  =  smx. 
I  tan  xdx  =  —  log  cos  x ;     i  ctn  xdx  =  log  si 
I  sec^ajc^o;  =  tancc;     / 


sina;. 


'.dx  =■  tan X ;     I  csc^icc^o;  =  —  ctn x. 


i.    I  coshccc^o;  =  sinho";     /  sinh  ic  c^a?  =  cosh  cc. 
I  tanh  xdx  =  log  cosh  cc  ;     /  ctnh  x  =  log  sinh 

®*    /  "Tl — s  =  -tan-M-)j  or ctn-M-)- 

,  J  d  +  x^      a  \a/  a  \a/ 

/dx          1              /x\           1          (X  -h  ic 
-^ 5  =  -  tanh-M  -  1  j  or  77-  log 
a^  —  x^      a             \a/         2  a     °a  —  x 

/dx               1    ^  ,     -  /x\          1   -      X  —  a 
-i -9.  = ctnh-^   -  )>  or  ~—  log • 
x^  —  a*          a              \a/        2a     ^  x-\-a 

3 

857200 


9.    f_^==sin-(^), 

...  J,  VaV-^.  . ;  :   V«/ 


FUNDAMENTAL  EQUATIONS 

©■ 


or  —  COS" 


'^^'    "'=log(a^,+  V^^T^). 


/c?x  1    '     Ja\ 
=-cos-M-)- 

r /^^     =  _  1  i.^/ci + v^^±^\ 

J  x^a'±x^         ^       \  X  / 

10.  r_^ 


a  \       —  a 


—  2 

or  — ;p=:tanh" 


1     f^ 


—J    or    — pctnh- 


^Af 


4-^a; 


11.  e^  =  cos  X  +  i  sin  cc ;     e""^  =  cos  x  —  i  sin  a;. 

12.  sinh  x  =  i(e'  -  e"^)  ;     cosh  a;  =  |(e^  +  e"^). 

13.  sin  xi  =  i  sinh  a? ;     cos  xi  =  cosh  x. 

14.  sin  X  =  —  i  sinh  xi ;     cos  x  =  cosh  xi. 

15.  log  u  =  log  (c2^)  —  log  c. 

16.  log  cc  =  log  (-  x)  +  (2k  4- 1)  TTi ;     log.x  =  (2.3025851)  •  log^^x. 

17.  log  (x  +  yi)  =  log  r  -\-  <f>i,     x  —  r  cos  <^,     ?/  =  r  sin  <^. 

sin  (x  +  yi)  =  sin  x  cosh  y  -^r  i  cos  x  sinh  ?/ ;  ^^^^ 

cos  (x  +  yv)  =  cos  X  cosh  ?/  —  *  sin  x  sinh  2/.  ^^ 


18.  sin-^x  4-  cos-^x  =  — ; 


tan~^x  +  ctn~^x 


19.  sinh-^x  =  log(x  +  Vx'-^H-l);        cosh-^x  =  log(x  +  Vx^— l); 


tanh~^x  =  \  log ; 

J.  —  X 


ctnh~-^x  =  -J-  log ; 

X  —  -L 


sech~^x  =  log 


1+ VT 


csch'^^x  =  log 


+vr+ 


20.  ctnh"-^x  =  tanh~-^x  -\-  \iri  -\-  kiri 


EATIONAL  ALGEBRAIC  FUNCTIONS  5 

^  I.   RATIONAL  ALGEBRAIC  FUNCTIONS 

A.   Expressions  involving  (a  -f-  hx) 
The  substitution  of  y  or  «  for  x,  where  y  —  xz  =  a  -{-  bx,  gives 

22.  {  x{a  +  bxydx  =  -  ly'^{y  -  a)dy. 

23.  J  x^  (a  4-  ^'a^)-c^a^  =  -^ij^  (^  -  ^T^^- 

2^     r     x-dx       ^    1      ny-aydy 
J  {a-^bxy      b^'  +  ^J  y"" 

25   r ^ = i_r(i^^:^ 

»  '  J  x"" (a  +  bx)""  a'^+^-^J  «»* 


'df« 


Whence 


dA.  27     f— ^^— =--^— . 

28     C      ^^       = 1 

J  {a  +  bxf  2b(a-\-bxy' 


/xdx         1 
^IfT^  =  ^[«  +  ^^  -  alog(«^  +  bx)l 


6  .  EATIONAL  ALGEBRAIC  FUNCTIONS 

J  (a-h  bxf      b^l     a-\-bx^  2{a  +  bxf\' 

/a?dx        in  1 

33. 


/x^dx  If  a^     1 

^^^^,  =  -^\a  +  bx-2a  log(«  +  to)  -  ^^^J^ 

/dx _1 
x{a-{-bx)~'~  a^^ 


g^     ,         --  1 ,      a  +  Sa; 


35  C      ^^       —        1        _  i_ ,     «_+ 

J  x(a  +  bxf  ^  a(a  +  bx)~  a"-  ^^      x 

36  C        ^^         —      i  _L.  ^  1      ^  +  ^^ 

J  x^{a-\-bx)  ax      a^     ^      x 


B.   Expressions  involving  (a  +  5af) 
^Q     r    dx  1  c  +  ic       r    dx  1         a;  —  c 

ot^      C     dx  1  x^fab 

J  a-\-bx^      -slab  a 

At\      r     ^^  1        ,      a-\-x^  —  ab 

J  a  +  &c2      2^-ab     ""  a-x-yJ-ab 


1       ,      ,     ,  05 V—  ab  1         ^  ,     .x^—ah 

or  .         tann~  ^ j  or      .         ctnn~  ^ • 

V— a6  «  V— a5  « 


*  j  {a  +  ^>ic2)2  ■"  2  a(a  H-  bx^)      2^  J  ^ 


^ 


RATIONAL  ALGEBRAIC  FUNCTIONS 

r  dx 1 X  2m  —  1   r        dx 

^'  J   (S^  +  bx'y^'^  ~2ma(a-h  bx'Y         2ma   J  (a-{-  bx^ 

,      r        xdx         _  1  /^         ^^  r    —    2T 

J  {a-^-bx'y^^  ~  2J  (a  +  bzY^^'  Iz  =  x  \. 

AK    r ___dx__^ _  _1_ ,     ^ 

•  J  x{a  +  bx')  ~  2^  ^^  a  4- 
J  a-^-bx"  ~b~bj  a  +  bx^ 


43. 


44. 


dx 


-^-bay" 

dx 


*  ^  ar^(a  +  ftar^)  ax      aj  a 

48  r   ^'^^    ^     -^      I  ^  r 

49     r  dx  _  1  r_^_dx b   r  dx 

^T     ^cx      C         dx  1   ,  iC" 

52.    I  =  —  loar • 

J  x(a  +  bx"")       an     ^  a  -\-bx'' 

r  dx  _  1  r        dx         _  b_  r        x'^dx 

•  J  (a  +  ^aj«)'»+i  ~"  aJ  (a  H-  te")"*      a  J  (a  +  ^>ir")'"+i ' 

r       x'^dx         _  1  r      g;"*""            a  ^     cc^'-^tZa; 
55     C  ^^  -  1  r  ^^ ^  r c^a; 


RATIONAL  ALGEBKAIC  FUNCTIONS 


-i 

a. 
+ 


■I 

+ 


+ 


+ 


+ 

a. 

IT 
+ 


+ 

a. 

IT 

+ 


g 

g 

g 

«   , 

,      ^       , 

,   s    ,    , 

1 

^.^ 

.--^ 

i^ 

1^ 

-11     - 

+ 

H 

+ 

tH 

+ 

v___ 

►C 

00 

O 

I— ( 

W 

W 
X 


O 


+ 

+ 
11 

t 


I 


I" 


I 

I 


Ch 


EATIONAL  ALGEBRAIC  FUNCTIONS 


|Mb        -_     Cxdx       1  1       ,.        h    Cdx 


--     Cxdx  bx  -\-  2  a      b  Ti 

«3j^= — ^      -J 


r  xdx  _      2a-^bx  _  b(2n-l)  Cdx 
J  Z«+i~  nqX""  nq        J  X"' 

^_     f  x%        X        h    .      ^^  b^-2ac  rdx 

^^     r^    7         (b^-2ac)x-{-ab   ,   2a  rdx 

66.    l—^dx  =  ^ ^ 1 I  — • 

J  X^  cqX  q  J   X 

^«     Cx'^dx  _      g;"'-^ n  —  m-\-l      b  Cx'^-'^dx 

J  JC»  +  i"~(2n-m-f-l)c^"~27i-m  +  l     cj     Z«+i 

m  —  1  a  rx"*-?dx 

2n-m-\-l'~cJ     Z'»+i 

^o     f  (^£c         1   ,      ar^        b     rdx 

„     f  ^a^  5.x        1  ^/  b'        c\  rdx 

70     f— ^—  - ?^ n-{-m  —  l     b^  r       dx 


^m-l^»+l 


m-1  a  /  a;'»-2jc'»+i' 


10  RATIONAL  ALGEBRAIC  FUNCTIONS 

D.   Rational  Fractions 
Every  proper  fraction  can  be  represented  by  the  general  form ; 

f{x)  ^  g^x-^  +  g^x-^-{-  g^x-^+  -"+9, 
F(x)  a:"  +  A;jCc«-i  +  k^x""-^  H \-k^  ' 

If  a,  b,  c,  etc.  are  the  roots  of  the  equation  F(x)  =  0,  so  that 

F(x)  =  (x  —  ay  (x  —  by  (x  —  cy---, 

fix)            A,                   A^  A^  A 

then     ^774  =  7 ^  +  7 '-r—:-^.        \.   .  +  -"+       ' 


F(x)       (x-ay      (x-ay-'^       (x-ay-^  x-a 

4-       ^^       + ^ + ^ +  -.-+-^ 

^  (x-  by  ^  (x-  by-''    (x-  by-^  ^     ^  x-b 
c,  c„  c,  c\ 

+  7 -Tr-^7 V^  +  7 V^  +  ---  + 


(x  —  cy     (x  —  cy~''     (x  —  cy~'^            x  —  c 
+ , 

where  the  numerators  of  the  separate  fractions  are  constants. 
If  a,  b,  Cy  etc.  are  single  roots,  then  ^=g'  =  r=---=l,  and 

F(x)      x  —  a      X  —  b      X  —  c 

The  simpler  fractions,  into  which  the  original  fraction  is  thus 
divided,  may  be  integrated  by  means  of  the  following  formulas : 

„       r     hdx       __  rhd(mx-\-n)  _  h  _^^ 

J  (mx -\- ny     J   m(mx -\- ny       m(l  — l)(mx -j- ny~^  i^ 

„ek      r  hdx  h  .       , 

72.  I  — —  loo:  (mx-^n). 

J  mx-\-n      m     ^^  ^ 

If  any  of  the  roots  of  the  equation  f(x)  =  0  are  imaginary,  the 
parts  of  the  integral  which  arise  from  conjugate  roots  can  be  com- 
bined, and  the  integral  thus  brought  into  a  real  form.  The  following 
formula,  in  which  i  =  V—  1,  is  often  useful  in  combining  logarithms 
of  conjugate  complex  quantities  : 

73.  log  (x  +  yi)  =  log  ?'  +  <^^,     x  =  r  cos  <^,     y  =  r  sin  <^. 


IRRATIONAL  ALGEBRAIC  FUNCTIONS  11 

^1|  II.   IRRATIONAL  ALGEBRAIC  FUNCTIONS 


A.   Expressions  involving  ^a-\-hx 


The  substitution  of  a  new  variable  of  integration,  y  =  V a  -f-  bx, 
gives 

74.  j-y/a  -f  bxdx  =  —  V(a  +  bxf. 

75.  r.V^TS<^x=-2(2^^1M^5i±M. 
J  15  b^ 


^  105  b^ 


77.  rv£±jg^x=2v;^Tto+ar  ,jf_- 

J         X  J  X  Va  4-  ^ic 

r__dx___  _  2  V(x  +  ^i^ 
J    y/a-j-bx  ^ 

rxd^2(2a-bx)^y-— 
80.    /      ,  =  -^^ TTTF^ Va  +  bx. 


81.      /   7=  =  — r^log     -^=^= j=)' 

Jx\a-{-bx       Va        \^a -\-bx -\-^a/ 


.    r       J^  =  ^tanh-  J^±^,  or  ^ctnh- J^±:^. 
J  x^a-\-bx       ■\la  y      a  Va  >      a 


82 

'    + 

,  dx  Va  +  bx        b     f*        dx 

83. 


/dx  _  _  Vq^  +  bx        b     r        di 

x^ Va  +  bx  <^^  2a J  xVa 


^bx 


12  IRRATIONAL  ALGEBRAIC  FUNCTIONS 


4- to 
c?a;  _  _      V<^  +  ^a;  {2n  —  Z)h    r  dx 


r         dx  _  _      Va  +  6a;      _  (^Zn—'S)})    r dx^ 

J  x«  Va  +  to  ~       (7i-l)aa;"-i       (2n-2)aJ  ^^-i-y/^ 

n  -  w-2 

88.    I  ^— -^- — ^^ =  b  j  (a  +  bx)  2   dx  +  a  j  ^'-—^ — ^ d'o;. 

^  x(a-\-bxY  J  x(a  +  bx)   ^  J   { 


+  to 
2 


(a  4-  to)2  c/  a;(o^  +  to)   2  ^   (<^  +  to)2 


B.   Expressions  involving  v  a;^  ±  d^  and  Va^  —  x^ 

90.  C-\/x'±a^dx  =  -^[ir  Vic^  ±  ci'^  ±  a^  \og{x  +  V^2^~^)]-* 

91.  r  Va^  -  x^dx  =  i  L  Va^  -  x^  +  «'  sin-i  (^^1  • 

92.  r^=^  =  log(x4-V^^T^^).^ 

93.  r    /^        =  sin-i(^-\  or  -  cos-^/^-). 

94.  / = -cos~M-)?  or -sec~M-)- 


95 
96 


IRRATIONAL  ALGEBRAIC  FUNCTIONS  13 


^1    •  97.    /  dx  =  Vic^  —  o?  —  a  cos~^  - 

J  X  X 


^^     .       xdx 
99 


100.  Cx  -slx^  ±  a^dx  =  ^  V(x2  ±  ^2)8^ 

101.  fic  Va^  -  a^^t^a;  =  -  ^  V(a2  _  x'f. 

102.  fV(^T^'dx 

103.  C-V^^F^^^^dx 
J  [.  V(^^3^«  +  ^  V^^3T^  +  ^  sin- g. 


104.  r_^=— ^£= 

^r^„  r  dx  X 

105.    /      .  = , 

xdx  —  1 


107. 
108 


xdx  1 


.    fic V(ic2  _j_  ^2>)8^^  ^  ^ V(a;2  ±  ay. 
109.  jx^(a^  -  a^ydx  =  -  ^  V(a2  _  x'f. 


*  See  note  on  page  12 


14  IREATIONAL  ALGEBRAIC  FUNCTIONS 

110.  jx^^x"  ±  a^dx 

111.  ixWa^-a^dx 

=  -^-s/(a'-xy  +  ^(xV^'^^'+a'sin-^^' 


113.  r-^2^=-^v^rr^  +  ^%m 


114.    /     .    .Ill— =T    -'"^^"^ 


2  '  a-iC 


115.    I  — —==  = ^ 


ii/.    I  ^ o(ic  = sm~^-- 

J  x^  X  a 

119.    r_-£^=.  =  -^=_sin-iE. 


C.   Expressions  involving  Va  +  h^+c^ 

4c 

Let    X  =  a  +  Z>ic  +  cx^,    q  =  A:ac  —  h^,    and    A;  = In    order 

to  rationalize  the  function  f{x,  Va  ■\-bx-\-  cx^)  we  may  put 
Vo-  4-  ftic  +  cx^  =  V±  c  vCr  +  ^B^i^,  according  as  c  is  positive 
or  negative,  and  then  substitute  for  x  a  new  variable  z,  such  that 

«  See  note  on  page  12 


IRRATIONAL  ALGEBRAIC    FUNCTIONS 


15 


z  =  V^  -\-Bx^-x^  —  ic,  if  c  >  0 ; 


z  = 5  if  c  <  0  and >  0 

X  —  c 


X  —  B 
A  f  wliere  a  and  yS  are  the  roots  of  the  equation 

A  -\- Bx  -  x^  =  0,  ii  c<0  SLud-^  <0. 

—  c 

By  rationalization,  or  by  the  aid  of  reduction  formulas,  may  be  ob- 
tained the  values  of  the  following  integrals  : 

or  —rsinh-M— =4=),  if  c>0. 

Vc  \-s/4:ac-by 

21.    f -7=^  = -7=  sm-M     .  -),iic<0. 


^ 


22 
23 
24 
25, 


dx     _2(2cx  +  h) 
XVX  qVx 

dx      _2(2cx-\-b)  /I 


r     dx  2{2cx-^-b)/l  \ 

•jx^Vx~    SqV^l    Kx^^^r 

r    dx      _  2{2cx  +  b)-\fx      2  7c(n-l)  C d 

'  J  X-^X~    {2n-l)qX-    ^     271-1    j  A— 1 


vz 


dx 


J  4(»  +  l)<;        ^2{n  +  l)kJ 

fxdx  _  Vx        b    r  dx 


dx 

Vx' 


'X'^dx 


6  IRRATIONAL  ALGEBRAIC  FUNCTIONS 

/xdx    __  2{bx-\-2  a) 
X^X~  q-y/X 

/xdx Vx  h     r     dx 

XWX~~  (2n-l)cX'^~YcJ  x-Vx' 

^^     rx'dx       I  X        ^b\    ,-  ,   3Z»2_4ac  r  dx 

r  x^dx   _  (2b''-4:ac)x-^2ab      1   r  dx 
J  X-y/x~  cqVx  cj   VX 

r  x^dx    _  (2b^-4:ac)x-^2ab      4.ac  +  (2n-S)b''  r      dx 
'  J  X''y/X~  (2n-l)cqX"-Wx  {2n-l)cq      J  x^-^Vx 

rxHx       (x"       5bx      5b^      ^^\VT      (^^      ^^M  r  ^^ 
^^•JVz~V3c      12c2"^8c«     36-V  Uc^      16cVj  Vx* 


39 


CxX'^dx  _     X"  Vx  ^     rX"c?a; 

•j     VX    "(271  +  1)0      2  J    VX* 

40     r^!^™^  ^    a;X"Vx   _  (2n  +  3)/>  T^X"^ 
•j      Vx         2(71 +  l)c       4(7i  +  l)cj     Vx 

~2(7i  +  l)cj    Vx* 
4l.ja^VX(^a:      V  8c  ^48c2      3c/    5o 


(l$-S)/^- 


IRRATIONAL  ALGEBRAIC  FUNCTIONS  17 

^    142.    / — ^= plogi +  — 7:=),ifa>0. 

.^o     r    dx  1        .   _,/    bx-\-2a    \ 

143.  /  7==     ; sm-M .  ?ifa<0. 

J  a:  VX       V-  a  \a:  V 6^  -  4  ac/ 

144.  /  —  = — ,  if  a  =  0. 

J  xVx  ^^ 

i  C   ^^    -      v^      ,  1  C    dx b_  r  dx 

J  xX""  Vx  ~  (2  71  -  1)  aX"      G^ J  ^x"-i  Vx      2  aj  a'»  Vz 
1     f    dx     _      Vx        b     r   dx 


145. 


xVx 

'y/Xdx  rz:   .    h    r  dx     .        C    dx 


/X'^dx X"  rx'^-Hx     ^  rx^'-Hx 

xVx~  {2n-l)-yfx      ^J    xVx       2J      ^x    ' 

iAe%      r^^dx  Vx       b    r    dx       ,        T  d^O! 

/x'^dx    _  1   rjf^2^^dx_  _  ^  rx^'-^dx       a  Px'^-^dx 
A^«Vx~  V  j^«-iVa     V  z^Va     V  jt"VA 

151     r^;!£!^  _  a^^-^A^Vx  _  (2n  +  2m-l)b  Cx'^-^X^dx 
Ak         *  J      Va     ~"  (2  7i  +  m)c  2c(2  7i4-m)     J        Va 

{m  —  l)a    rx'^-^X'^dx 
(2n-\-m)cJ        Va 

152.    ^      <"-  ^ 


'h 


'A"  Va  (m  —  1)  ax*"  - 1  A" 

(2n+2m-S)b  C         dx 
2a(m-l)      J  x'^-^X'^Vx 

(2n-\-7n  —  2)c  r         dx     • 
(m-l)a      J  x^-^X^-\/x' 


18  IREATIONAL  ALGEBRAIC  FUNCTIONS 


153     r  X"dx   _        X-Wx         (2n-l)h  rX-2^ 
'  J  x^Vx  (m-l)a^— 1       2(m-l)  J  ^m-iVA 

.         (2n-l)c  r  X^'-^dx 
m-1    J  X'^-^Vx 

J  (a'  +  b'x)  Vx       V-  A  2  5'  V-  A  A' 

or  JLin.2A4-m(a^  +  ^>^^)-2^>^VXx 

where      m  =  5^'  —  2  a'c  and  /?.  =  a^*'^  —  aW  +  m'^. 

If  7i  =  0,  the  value  of  the  integral  is  -  2  5' VA/[m(a'+ J'cc)]. 

D.   Miscellaneous  Algebraic  Expressions 

155.  /  ■\/2 ax  —  x^  dx  =  ^l{x  —  a)  V2 ax  —  x""  +  a^ sin-i(aj  —  a)/a]. 

156.  r-=£=  =  cos-(^:if). 
J   ■\/2ax-x'  \     a     / 


,^^     C dx __2_,      _,       -h\a  +  hx) 

157.    /      ,  ,  =     / :  tan-1  *         ^ ^ 

J   -ya  -\-  hx  .  Va'  +  ^>'a:;       a 

2     ^     ,  _,     I  //(^  +  hx) 
or  — 7=tanh     \  ttVtttt 


+  ^»£C  .  Va'  +  ^»':z;       V^^   ^^      N  ^»  (a'  +  h'x) 


158.    C^{a  +  ^'a^)(a'  +  b^x)  dx  =  ^  "^  ^  ^^^^f  "^  ^'^  V(a  +  hx){a^  +  b'x) 
'J   -Ja^hx.  ^a'  -{-b^x 


Sbb 
159 


r     \a'-hb'x       _  Vg  +  bx  •  ^a'  +  h'x       k    r  dx 

J    \a  +  bx    '^~  b  2bJ   ^a  +  bx-\/a'-{-b'x' 


160.   f\l^-^  dx  =  sin-la^  -  Vl  -  x^ 


IRRATIONAL  ALGEBRAIC  FUNCTIONS 


19 


■^/(x-a){a'-x)  ya'-a 

jgg      r (px-\-q)dx  ^  q-\-a'p   P dx 

J  {x-a%x-h')^a  +  hx  +  cx'       (^'  -  b'  J  (x-a')-\/a-{-bx-\-cx' 


dx 


q-\-b'p  r 
a'  -  b'  J  (x  -  b')-y/a  -\- bx -^  ex" 


164 


cx^ 


/dx 
(a'-\-b'x)-\/a-^bx-{- 

1       ,       /2  ^  +  m(a^  +  b'x)-2b'  •\lh{a  -\-bx-\-  c^\ 


or 
where 


"I^  \^2 V  sf—hia  +  bx-^cx^)) 


m  =  bb'  ^2  a'c  and  h  =  a&'^  —  a'bb'  +  ca 


165.    / ,  =--;\~i r-tan-iic\-7- — ; — ^j 

J  (a'  +  c'x^)  Va  +  cx^      (^   ^  «^c  -  « ^^  ^  ^  («^  +  ^»  ) 


^ 


or 


a'  ^  a'c  —  ac'        Va  +  cx^  —  x yJia'c  —  aG')/a' 


C              xdx                    1            c^        ^       1  ^  /^  (^  +  ^*^ 
166.    I ,  =iA/~l :tan-iV-^^ fr 

J_    I       g'  Va  +  Gx^--yl{ac'  -  6^'c)/c' 

2  c'^/  o^c'  -  a'c  ^^  Va  +  cx^  +  V(ac'  -  a'c)/c'' 


20  TRANSCENDENTAL  FUNCTIONS 

III.   TRANSCENDENTAL  FUNCTIONS 

67.  I  sin  xdx  —  —  cos  x. 

68.  I  ^\Ti?xdx  =  —  -J- cos X  sin x  -\-  ix  =  \x  —  ^  sin  2 x. 

69.  I  s,m^xdx  =  —  -J  cos  ic (sin^ic  -f  2).        . 

„n     r  '         7  sin"-^a;cos£c  ,  n  —  1  T  .       „     , 

70.  I  sin"£crfa7  = 1 I  sin"-^^^^,^ 

J  n  n    J 

71.  I  cosccc^ic  =  sinic. 

72.  r  cos^a;c?ic  =  ^  sin  cc  cos  cc  +  ^  a^  =  ^  cc  -|-  j-  sin  2  ic. 

73.  I  cos^xdx  —  -J  since  (cosmic  +  2). 

74.  I  cos"icc?£c  =  -  cos^'^ic  sin  a;  H /  (iO%*'~'^xdx. 

J  n  n    J 

75.  I  sin  X  cos  xdx  =  ^  sin^ic. 

76.  I  sin^ic  co^^xdx  =  —  ^ (J  sin  4 ic  —  £c). 

»»     r  '  m    ^  cos'«+ix 

77.  I  sin  X  cos"*ic  ace  = — ^  • 

»a     r  •   m  1        sin'^+^ic 

78.  I  sin"*ic  cos  xdx  — —  • 

J  ^  +  1 

ffft     r      m      •  «     7        cos'^-^icsin^+iic  ,   m  —  1   r      ^  „     •  ,     7 

79.  I  cos"*cc  sin**icaic  = h I  cos"*~^£c  sm^icaa;. 

J  m  -\-  n  m-h  /ij 

on     r      m     •  n    J           sin"-^iccos'»+^a!  ,  71  — 1    T      ^     •  „  2     7 
oO.    I  GOS"*xsin''xdx  = 1 |  cos'"a!Sin"^xaa:i 

«-      feos'^xdx  _  cos"'''"^a;  m  —  n  -\- 2  Cco^'^xdx 

J      sin'a;  (ti  —  l)sin"~^ic  ti  — 1     J    sin''~'*a; 


TRANSCENDENTAL  FUNCTIONS  21 

~^xdx 


mL        _      rcos^'xdx  _         cos"*~^a;  m  —  1    rcos'^-^x 

^  J      sin"ic         (m  —  ?i)  sin"~^ic      m  —  nj        sin^a; 


sm' 


(i-) 


sin"'x  cos"a; 


n 


J^ 1 m  +  7i-2  r  cga; 

—  1    sin'^-^x  •  cos"~^a7  ti  —  1     J  sin"*ic  •  cos^-^a? 

71-2  r 
-1    js 


1  1  7n-{-n  —  2  f*  dx 


m  — 1   sin"*~^a;  •  cos^'^a;  w  — 1    J  sin"*' ^£c- cos" a;. 


/-; — ^^^^^^ =  log  tan  X. 
sin  a;  cos  a; 

J.-     r_dx^_  _  _      1  cos  a;         m  —  2  T 

'  J  sin'"a;  m  —  1    sin"'~^a7      m  —  ij  si 


sm"*  "a; 


QR     r  ^^     —      ^  sina;         n  —  2   T     c?a; 

J  cos^x      71  —  1   cos"-^a;      n  —  lj  cos^-^a; 

87.  I  tan  xdx  =—  log  cos  a;. 

88.  I  tan^ ajc^aj  =  tana;  —  ar. 

/tan**~^a!        /* 
tan"a;c?a;  = t-  —  j  t^n^'-^xdx. 

90.  I  ctnajc^a;  =  log  sina;. 

91.  I  ctn^ajc^a;  =  —  ctn  x  —  x. 

/ctn'*~^a;        r 
ctn"a'6?x  = I  ctn" -2  a;  c^a;. 

93.    I  seca;6?a;  =  logtan  (-J  + -)• 


94.    I  sec^ajc^a;  =  tana;. 


22  TRANSCENDENTAL  FUNCTIONS 

195.    lsec''xdx=    I — ;; —  196.    I  cscsc^^a;  =  losr  tan^a;. 

197.    I  csc^jcc^ic  =— etna?.  198.    /csc"icc?x=    /    .  ^    • 

J  J  J  sm"a: 

199.    / — r^ =  — 7==tan-i — — ^— , 

J  a-{-bcosx      -yJd^  _  ^2  a +  6 


1        ,      V^^  —  a^  tan  hx-\-a-\-'b 

.  or      ■  log     ,  ^ —  3 

t=  -sly" -a?     ^  ■\/lP--a^t^xv\x-a-h 

^  2       ,     ^   ,  V^^H^tan^a; 

.,  or      ,  tann~^ — ^— > 

V  V^^  -  a2  a  +  ^> 

t=  / 

I  2         ,    ,    ,  V  ft^  —  a^  tan  -i-  a; 

'  or  — ==ctnh-i — - — 


200.     (   ——J ; : 

J   a  -\-  0  cos  X  -\-  c  sin  x 


.(a  —  b)  tan  i  x  +  c 

, tan-^  ^^ ,   ^         ^     ^—: 

Va2  _  J2  _  ^2  Va^  -b^-c" 


t=                      1            ,      (a  -  5)  tan  4a;  +  c  -  VZ»24-  c^-  a^ 
x/     or      ,  =  log ^ , 

V  V^2_^c2-a2     ^(a_^,)tan|a;  +  c  +  V^2:^7-^ 

V  —  2  .     1    1  («^  —  ^)  tan  4-  a?  +  c 

^  tanh-^  ^ ,  ^         ^     ^— > 

—  2  ,   ,    1  (a  —  ^)  tan  4-  a?  +  c 

V^TfTTT^  V62  +  c^  -  a^ 

201.  /  x  sin  ajcZa?  =  sin  x  —  x  cos  a?. 

202.  /  x^  sin  asc^a;  =  2  a;  sin  a;  —  (a:*^  —  2)  cos  a;. 

203.  J  x^sinxdx  =  (Sx^—  6)  sin  a;  —  (a;*  —  6  a;)  cos  x. 

204.  I  a:"*  sin xdx  =  —  ^'"cos  x -\- m  I  x"*" ^cos  a;c?x. 


TEANSCENDENTAL  FUNCTIONS  23 

^H      205.    /  X  cos  xdx  =  cos  x  -[-x  sin  x. 

206.  I  cc^cos  xdx  =  2  X  Go^  X  -{-  (x^  —  T) sin  ic. 

207.  /  cc'cos  £C(^ic  =  (3  cc^  —  6) cos  x  -\-(x^  —  6  x)  since. 

208.  I  ic"* cos icc?x  =  cc"" sin a^  —  m  I  o-"""^ since fZa;. 

---      /*sinic  ,  1        since  1        /^coscc  ^ 

209.  I  dx= ' 7  H I  7  dx. 

J     x""  m  —  1    £c"*-i      m  —  IJ  cc"*-' 

«,-      Tcoscc  ,  1        coscc  1        Tsincc  , 

210.  I  dx  = 7   I  r  dx. 

J     x^  m  —  1    cc"*~^       m  —  1  J   £c"'~^ 

on      r  since       _  cc^  x^ cc^  cr;^ 

213.  I  sin  (mx  +  a)  •  sin  (nx  +  &)  c?£c 

_  sin  (mx  —  nx  -\-  a  —  b)       sin  (mcc  -\-  nx  -{-  a  -\-  b) 

2  (m  —  n)  2(m  ■\-  n) 

214.  I  cos  (mx  +  a)  •  cos  (nx  -\-  b)dx 

_  sin  (mx  ■\-  nx  ■\-  a  -\- b^       sin  (mcc  —  rice  -f-  a  —  J) 

~  2  (m  4-  w)  2  (m  -  ti) 

215.  I  sin  (mx  -\-  a)  •  cos  (nx  -\-  b)dx 

_      cos  (mx  -\- nx -\-  a -{- b)      cos  (ma?  —  nx  -^  a  —b) 
~'~  2(m  +  n)  ~  2(m  —  n) 

216.  I  sin  (mcc  +  a)  •  sin  (mcc  +  ^)  c?cc 

X          ,,         ^       sin  (mcc  +  a)  •  cos  (mcc  +  5) 
=  -  •  cos  (Z>  -  a) ^ ^ ^ ^• 

2  ^  -^  2m 

217.  I  sin  (mcc  +  a)  •  cos  (mcc  -^b)dx 

sin  (mcc  -f  a)  •,  sin  (mcc  +  5)      cc     .     ,, 

= ^ '- ^ ^  -  -  •  sm  (^  -  a). 

2m  2         ^  ^ 


24  TRANSCENDENTAL  FUNCTIONS 

218.  /  cos  {mx  H-  a)  •  cos  (mx  +  b)dx 

X  „         ^       sin  (mx  -h  ci)  cos  (ttix  +  &) 

=  -  •  cos  (^»  -  a)  H ^ :^ ^ ^ 

2  ^  ^  2m 

219.  I  sin~^ icc?a;  =  x  siii~^ic  +  Vl  —  x^, 

220.  I  cos~^icc?ic  =  X  cos~^£c  —  Vl  —  x^. 


221 
222 


I  taii-^a;(^x  =  ic  tan-^£c  —  ^  log  (1  4- ar^. 
I  atn-'^xdx  =  x  ctn-^a;  +  ^  log  (1  +  a;^. 

223.  /  versin"^a;c?ic  =  (a;  —  1)  versin~^ic  +  V2ic  —  cc^. 

224.  C(sm-^xydx  =  ic  (sin-icc)^  -  2  x  +  2  Vl  -  cc^  gin-ia;. 

225.  fa; .  siii-^xdx  =  ^  [(2  cc^  _  l)  sin-iic  -f  a;  Vl  -  x^]. 

^^^     r       .17        x'^+^sin-ia;  1        r  x^'+^dx 

226.  I  a;"sin-ix<^a;  = — ^ — r  I  ■   , 

J  n  +  1  ^  +  lJVl-ar* 

_.^     r  1     7        a;"+icos-ia^   ,       1        f  a;"+^c?a^ 

227.  ./  a;"cos-ix^x  = — j —  +  — — r  |     /:, -' 

J  7^  +  1  n-{-lJ   Wl  —  x^ 

^^^     r     .       17        a;"+Uan-ia;  1        Px^'+^dx 

228.  I  a;"  tan-i xc^a;  = — j — r  |   .   .     »  - 

J  71  +  1  ^  +  1J    1  +  ar^ 

229.  I  logajc^a;  =  a;  log  a;  —  x. 

230.  r2^^^^x  =  -^aoga^)"+^ 

J  X  71  +  1    ^  ^ 


231.    j— ^^  =  log(loga^). 


dx 

log  a; 


232 


r      dx 

J  x(\ogx) 


:y  (71-1)  (log  a;)"-i 

233.    f.T-loga^cZa;=:a;-+^r^^-  ,     ^.J. 


^ 


TRANSCENDENTAL  FUNCTIONS  25 


SA 


/" 


234.    I  e'^dx  =  - 


xe'^dx  =  —  iax-l). 


x'^e'^dx  = I  x"'-'^e"^dx. 

a  aj 

c^^c     C      ^  7        e"^ logic       1    re«^  , 

238.  I  e«^logicc?a;= ^ |  — dx. 

J  a  aJ    X 

239.  Je-  .  .inpxdx  =  ^-(^sin^^-^_^-|^cos^.)^ 

241.  I  sinh  ajc?ic  =  cosh  a;;     i  GO%h.xdx  =  sinhcc. 

242.  I  tanh  xdx  =  log  cosh  a? ;     /  ctnh  xdx  —  log  sinh  ic. 

243.  rsech£c^a;  =  2taii-i(e^). 

244.  I  cschicc^a;  =  logtanh(-)' 

245.  I  £c  ^mh.xdx  =  «  cosh  x  —  sinh  x. 

246.  I  x  cosh  xdx  =  x  sinh  ic  —  cosh  ic. 

247.  I  cosh^a;c?a;  =  ^  (sinhx  coshic  +  x). 

248.  I  sinh  x  cosh  £cc?a;  =  \  cosh  (2  cc). 

249.  /  sinh^  icc^ic  =  ^  (sinh  a;  cosh  x  —  x). 


26  MISCELLANEOUS  DEFINITE  INTEGRALS 

IV.   MISCELLANEOUS  DEFINITE  INTEGRALS    ^«g 

250.  j^*^:^  =  |'  if  «>0;  0,  if  a  =  0;  -|,  if  a<0. 

251.  r  ic''-ie-*c^x=  r  I  log- I      dx  =  T(n). 

r(7i  + 1)  =  n .  r(7i),  if  71  > 0.  r(2)  =  r(i)  =  i 

T(n-\-l)  =  nl,  if  71  is  an  integer.         r (^)  =  V^. 

r  (7.)  =  n  (7.  - 1).  z  (y)  =  i),  [log  r  (t/)] 

Z(l)=- 0.577216. 

oco      r'  ™    1/-.         Nn   1^  r*     x^'-'dx  T(m)T(n) 

Jo  ^  Jo    (l  +  ^r  +  '*      r(7^  +  7i) 

253.    J     sin"xe^ic  =  i     cos'^xdx 
Jo  Jo 

1  .  3  .  5  .  .  .  (71  -  1)     TT       .-         . 

=  — ^^ — - — ^ —    ,  ^  •  —  >  if  71  IS  an  even  integer ; 

^  •  4  •  O  •  •  •  ( 71-)  ^ 

2.4. 6-. .(7.-1)     ...  . .  _ 

=  -q — ^ — z — ^7 5  II  71  IS  an  odd  integer ; 

1  •  O  .  O  •    <    ...  71 


p/^  +  l 


1    /-    V     2    / 
=  71^'^  — 7 T  for  any  value  of  7i  greater  than  —1. 


rt_-      /**  sin77iccc?aj      tt..  /x/^.n  ^^         7r._ 

254.    /      2 =  77'  ifw>0;  0,  if7?i  =  0;  -->  if77i<0. 


X  2  '  "  '""^  " '  "'  "  '"^       ^ '       2 


nee        T "   sln  OJ  •  COS  77107  C?iC  ^      .„ 

255.    /      =  0,  if  771-  <  -  1  or  77i  >  1 ; 

Jo  ^ 

—  >  if7?i  =  —  1  or  771  =  1;  — >  if  —l<m<l. 
4  2 

oKfi      r*  sin^ajc?a;      tt    . 


MISCELLANEOUS  DEFINITE  INTEGRALS 


27 


in  257.    r"cos  {x'')dx  =  rsin  (x')dx  =  i  ^ 

TT 

2* 

258.    1     sin  kx  sin  mxdx  =  /     cos  kx  cos  mxdx  =  0,  [7c  =7^  m]. 

259.    /    sin  A;ic  cos  mxdx  =  -z ^>  if  A;  —  m  is  odd ; 

Jo                                    ^  -  ^ 

=  0,  if  A;  —  m  is  even. 

260.    1    sm^Tnxdx^  1    GOS^mxdx  =  '^' 

261.    /    sin  kx  cos  kxdx  =  0. 

263 


264. 


Jo    G^  +  ^cosa;       vV_^ 

Jf "  cosmxdx      TT       .  ,    .„       .     ^ 


Vl-A:^ 


:^  =  iir 


sm'ic 


Jr*  cosa;c?a;  _   T"  sinxdx  _      & 

265.  r 

^    '=fh©'»--(i-?--(SI)'--]'"- 

IT 


<1. 


266. 

267 
268 


Wl-k''sm''x'dx  =  i: 


.jrv.-^.=i:5^=^,ifn>-i,«>o. 


28           MISCELLAKEOUS  DEFINITE  INTEGRALS 
>'    \     e  "^    ^dx  = Yy >  if  a  >  0. 


270. 


271.    I     e-'^QQ^mxdx^-r-^ ^>ifa>0. 


£ 
£ 


7n 
272.    /     e-'^sin mxdx  =  -rr- z  j  if  a  >  0. 


r     -^ 

«-.«         r*  2  2  y        -.  V  TT  •   e     4a2 

273.    I     e-«^  COS  bxdx  = >  if  a  >  0. 

Jo  2a 

274.  rM^.z.=-^. 

^^^•i   1  +  0.^^-      12 

280.  jT'x-  logg)"<fa  =  (£^|^>  [»  + 1  >  0,  »  + 1  >  0]. 

'     log  sin  ictZic  =    I     log  COS  iPc^x  =  —  —  •  log  2. 

0  Jo  ^ 

XT  ^ 

ic  •  log  sin  iccZic  =  — —  log  2. 


TABLES 


29 


Natural  Logarithms  of  Numbers  between  1.0  and  9.9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1. 

0.000 

0.095 

0.182 

0.262 

0.336 

0.405 

0.470 

0.531 

0.588 

0.642 

2. 

0.693 

0.742 

0.788 

0.833 

0.875 

0.916 

0.956 

0.993 

1.030 

1.065 

3. 

1.099 

1.131 

1.163 

1.194 

1.224 

1.253 

1.281 

1.308 

1.335 

1.361 

4. 

1.386 

1.411 

1.435 

1.459 

1.482 

1.504 

1.526 

1.548 

1.569 

1.589 

5. 

1.609 

1.629 

1.649 

1.668 

1.686 

1.705 

1.723 

1.740 

1.758 

1.775 

6. 

1.792 

1.808 

1.825 

1.841 

1.856 

1.872 

1.887 

1.902 

1.917 

1.932 

7. 

1.946 

1.960 

1.974 

1.988 

2.001 

2.015 

2.028 

2.041 

2.054 

2.067 

8. 

2.079 

2.092 

2.104 

2.116 

2.128 

2.140 

2.152 

2.163 

2.175 

2.186 

9. 

2.197 

2.208 

2.21d 

2.230 

2.241 

2.251 

2.262 

2.272 

2.282 

2.293 

Natural  Logarithms  of  Whole  Numbers  from  10  to  109 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2.303 

2.398 

2.485 

2.565 

2.639 

2.708 

2.773 

2.833 

2.890 

2.944 

2 

2.996 

3.045 

3.091 

3.135 

3.178 

3.219 

3.258 

3.296 

3.332 

3.367 

3 

3.401 

3.434 

3.466 

3.497 

3.526 

3.555 

3.584 

3.611 

3.638 

3.664 

4 

3.689 

3.714 

3.738 

3.761 

3.784 

3.807 

3.829 

3.850 

3.871 

3.892 

5 

3.912 

3.932 

3.951 

3.970 

3.989 

4.007 

4.025 

4.043 

4.060 

4.078 

6 

4.094 

4.111 

4.127 

4.143 

4.159 

4.174 

4.190 

4.205 

4.220 

4.234 

7 

4.248 

4.263 

4.277 

4.290 

4.304 

4.317 

4.331 

4.344 

4.357 

4.369 

8 

4.382 

4.394 

4.407 

4.419 

4.431 

4.443 

4.454 

4.466 

4.477 

4.489 

9 

4.500 

4.511 

4.522 

4.533 

4.543 

4.554 

4.564 

4.575 

4.585 

4.595 

10 

4.605 

4.615 

4.625 

4.635 

4.644 

4.654 

4.663 

4.673 

4.682 

4.691 

Values  in  Circular  Measure  of  Angles  which  are  given  in 
Degrees  and  Minutes 


1' 

0.0003 

9' 

0.0026 

3° 

0.0524 

20° 

0.3491 

100° 

1.7453 

2' 

0.0006 

10' 

0.0029 

4° 

0.0698 

30° 

0.5236 

110° 

1.9199 

3' 

0.0009 

20' 

0.0058 

6° 

0.0873 

40° 

0.6981 

120° 

2.0944 

4' 

0.0012 

30' 

0.0087 

6° 

0.1047 

50° 

0.8727 

130° 

2.2689 

6' 

0.0015 

40' 

0.0116 

7° 

0.1222 

60° 

1.0472 

140° 

2.4435 

6' 

0.0017 

50' 

0.0145 

8° 

0.1396 

70° 

1.2217 

150° 

2.6180 

7' 

0.0020 

1' 

0.0175 

9° 

0.1571 

80° 

1.3963 

160° 

2.7925 

8' 

0.0023 

2' 

0.0349 

10" 

0.1745 

90° 

1.5708 

170° 

2.9671 

30 


TABLES 
Natural  Trigonometric  Functions 


Angle 

Sin 

Csc 

Tan 

Ctn 

Sec 

Cos 

0° 

0.000 

GO 

0.000 

00 

1.000 

1.000 

90° 

1 

0.017 

57.30 

0.017 

57.29 

1.000 

1.000 

89 

2 

0.035 

28.65 

0.035 

28.64 

1.001 

0.999 

88 

3 

0.052 

19.11 

0.052 

19.08 

1,001 

0.999 

87 

4 

0.070 

14.34 

0.070 

14.30 

1.002 

0.998 

86 

6° 

0.087 

11.47 

0.087 

11.43 

1.004 

0.996 

86° 

6 

0.105 

9.567 

0.105 

9.514 

1.006 

0.995 

84 

7 

0.122 

8.206 

0.123 

8.144 

1.008 

0.993 

83 

8 

0.139 

7.185 

0.141 

7.115 

1.010 

0.990 

82 

9 

0.156 

6.392 

0.158 

6.314 

1.012 

0.988 

81 

10° 

0.174 

5.759 

0.176 

5.671 

1.015 

0.985 

80° 

11 

0.191 

5.241 

0.194 

5.145 

1.019 

0.982 

79 

12 

0.208 

4.810 

0.213 

4.705 

1.022 

0.978 

78 

13 

0.225 

4.445 

0.231 

4.331 

1.026 

0.974 

77 

14 

0.242 

4.134 

0.249 

4.011 

1.031 

0.970 

76 

16° 

0.259 

3.864 

0.268 

3.732 

1.035 

0.966 

76° 

16 

0.276 

3.628 

0.287 

3.487 

1.040 

0.961 

74 

17 

0.292 

3.420 

0.306 

3.271 

1.046 

0.956 

73 

18 

0.309 

3.236 

0.325 

3.078 

1.051 

0.951 

72 

19 

0.326 

3.072 

0.344 

2.904 

1.058 

0.946 

71 

20° 

0.342 

2.924 

0.364 

2.747 

1.064 

0.940 

70° 

21 

0.358 

2.790 

0.384 

2.605 

1.071 

0.934 

69 

22 

0.375 

2.669 

0.404 

2.475 

1.079 

0.927 

68 

23 

0.391 

2.559 

0.424 

2.356 

1.086 

0.921 

67 

24 

0.407 

2.459 

0.445 

2.246 

1.095 

0.914 

66 

26° 

0.423 

2.366 

0.466 

2.145 

1.103 

0.906 

66° 

26 

0.438 

2.281 

0.488 

2.050 

1.113 

0.899 

64 

27 

0.454 

2.203 

0.510 

1.963 

1.122 

0.891 

63 

28 

0.469 

2.130 

0.532 

1.881 

1.133 

0.883 

62 

29 

0.485 

2.063 

0.554 

1.804 

1.143 

0.875 

61 

30° 

0.500 

2.000 

0.577 

1.732 

1.155 

0.866 

60° 

31 

0.515 

1.942 

0.601 

1.664 

1.167 

0.857 

69 

32 

0.530 

1.887 

0.625 

1.600 

1.179 

0.848 

68      . 

33 

0.545 

1.836 

0.649 

1.540 

1.192 

0.839 

67     i 

34 

0.559 

1.788 

0.675 

1.483 

1.206 

0.829 

66 

36° 

0.574 

1.743 

0.700 

1.428 

1.221 

0.819 

65° 

36 

0.588 

1.701 

0.727 

1.376 

1.236 

0.809 

64 

37 

0.602 

1.662 

0.754 

1.327 

1.252 

0.799 

63 

38 

0.616 

1.624 

0.781 

1.280 

1.269 

0.788 

62 

39 

0.629 

1.589 

0.810 

1.235 

1.287 

0.777 

61 

40° 

0.643 

1.556 

0.839 

1.192 

1.305 

0.766 

60° 

41 

0.656 

1.524 

0.869 

1.150 

1.325 

0.755 

49 

42 

0.669 

1.494 

0.900 

1.111 

1.346 

0.743 

48 

43 

0.682 

1.466 

0.933 

1.072 

1.367 

0.731 

47 

44 

0.695 

1.440 

0.966 

1.036 

1.390 

0.719 

46 

46° 

0.707 

1.414 

1.000 

1.000 

1.414 

0.707 

46° 

Cos 

Sec 

Ctn 

Tan 

Csc 

Sin 

Angle 

TABLES 


31 


Values  of  the  Complete  Elliptic  Integrals,  K  and  £,  for  Different 
Values  of  the  Modulus,  k 


-=f: 


dz 


Vl-A:2 


sm-'z 


sin-U- 

K 

E 

sin-U- 

K 

E 

sin-U- 

K 

E 

0° 

1.5708 

1.5708 

60° 

1.9356 

1.3055 

81.0° 

3.2553 

1.0338 

1° 

1.5709 

1.5707 

61° 

1.9539 

1.2963 

81.2° 

3.2771 

1.0326 

2° 

1.5713 

1.5703 

62° 

1.9729 

1.2870 

81.4° 

3.2995 

1.0313 

3° 

1.5719 

1.5697 

63° 

1.9927 

1.2776 

81.6° 

3.3223 

1.0302 

40 

1.5727 

1.5689 

64° 

2.0133 

1.2681 

81.8° 

3.3458 

1.0290 

6° 

1.5738 

1.5678 

66° 

2.0347 

1.2587 

82.0° 

3.3699 

1.0278 

6° 

1.5711 

1.5665 

66° 

2.0571 

1.2492 

82.2° 

3.3946 

1.0267 

70 

1.5767 

1.5649 

67° 

2.0804 

1.2397 

82.4° 

3.4199 

1.0256 

8° 

1.5785 

1.5632 

68° 

2.1047 

1.2301 

82.6° 

3.4460 

1.0245 

9° 

1.5805 

1.5611 

69° 

2.1300 

1.2206 

82.8° 

3.4728 

1.0234 

10° 

1.5828 

1.5589 

60° 

2.1565 

1.2111 

83.0° 

3.5004 

1.0223 

11° 

1.5854 

1.5564 

61° 

2.1842 

1.2015 

83.2° 

3.5288 

1.0213 

12° 

1.5882 

1.5537 

62° 

2.2132 

1.1921 

83.4° 

3.5581 

1.0202 

13° 

1.5913 

1.5507 

63° 

2.2435 

1.1826 

83.6° 

3.5884 

1.0192 

14° 

1.5946 

1.5476 

64° 

2.2754 

1.1732 

83.8° 

3.6196 

1.0182 

16° 

1.5981 

1.5442 

66° 

2.3088 

1.1638 

84.0° 

3.6519 

1.0172 

16° 

1.6020 

1.5405 

66.6° 

2.3261 

1.1592 

84.2° 

3.6853 

1.0163 

17° 

1.6061 

1.5367 

66.0° 

2.3439 

1.1546 

84.4° 

3.7198 

1.0153 

18° 

1.6105 

1.5326 

66.6° 

2.3622 

1.1499 

84.6° 

3.7557 

1.0144 

19° 

1.6151 

1.5283 

67.0° 

2.3809 

1.1454 

84.8° 

3.7930 

1.0135 

20° 

1.6200 

1.5238 

67.6° 

2.4001 

1.1408 

86.0° 

3.8317 

1.0127 

21° 

1.6252 

1.5191 

68.0° 

2.4198 

1.1.362 

86.2° 

3.8721 

1.0118 

22° 

1.63b7 

1.5141 

68.6° 

2.4401 

1.1317 

86.4° 

3.9142 

1.0110 

23° 

1.6365 

1.5090 

69.0° 

2.4610 

1.1273 

86.6° 

3.9583 

1.0102 

24° 

1.6426 

1.5037 

69.6° 

2.4825 

1.1228 

86.8° 

4.0044 

1.0094 

26° 

1.6490 

1.4981 

70.0° 

2.5046 

1.1184 

86.0° 

4.0528 

1.0087 

26° 

1.6557 

1.4924 

70.6° 

2.5273 

1.1140 

86.2° 

4.1037 

1.0079 

27° 

1.6627 

1.4864 

71.0° 

2.5507 

1.1096 

86.4° 

4.1574 

1.0072 

28° 

1.6701 

1.4803 

71.6° 

2.5749 

1.1053 

86.6° 

4.2142 

1.0065 

29° 

1.6777 

1.4740 

72.0° 

2.5998 

1.1011 

86.8° 

4.2744 

1.0059 

30° 

1.6858 

1.4675 

72.6° 

2.6256 

1.0968 

87.0° 

4.3387 

1.0053 

31° 

1.6941 

1.4608 

73.0° 

2.6521 

1.0927 

87.2° 

4.4073 

1.0047 

1    32° 

1.7028 

1.4539 

73.6° 

2.6796 

1.0885 

87.4° 

4.4812 

1.0041 

f    33° 

1.7119 

1.4469 

74.0° 

2.7081 

1.0844 

87.6° 

4.5619 

1.0036 

'     34° 

1.7214 

1.4397 

74.6° 

2.7375 

1.0804 

87.8° 

4.6477 

1.0031 

36° 

1.7312 

1.4323 

76.0° 

2.7681 

1.0764 

88.0° 

4.7427 

1.0026 

36° 

1.7415 

1.4248 

76.6° 

2.7998 

1.0725 

88.2° 

4.8479 

1.0022 

37° 

1.7522 

1.4171 

76.0° 

2.8327 

1.0686 

88.4° 

4.9654 

1.0017 

38° 

1.7633 

1.4092 

76.6° 

2.8669 

1.0648 

88.6° 

5.0988 

1.0014 

39° 

1.7748 

1.4013 

77.0° 

2.9026 

1.0611 

88.8° 

5.2527 

1.0010 

40° 

1.7868 

1.3931 

77.6° 

2.9397 

1.0574 

89.0° 

5.4349 

1.0008 

41° 

1.7992 

1.3849 

78.0° 

2.9786 

1.0538 

89.1° 

5.5402 

1.0006 

42° 

1.8122 

1.3765 

78.6° 

3.0192 

1.0502 

89.2° 

5.6579 

1.0005 

43° 

1.8256 

1.3680 

79.0° 

3.0617 

1.0468 

89.3° 

5.7914 

1.0005 

44° 

1.8396 

1.3594 

79.6° 

3.1064 

1.0434 

89.4° 

5.9455 

1.0003 

46° 

1.8541 

1.3506 

80.0° 

3.1534 

1.0401 

89.6° 

6.1278 

1.0002 

46° 

1.8691 

1.3418 

80.2° 

3.1729 

1.0388 

89.6° 

6.3504 

1.0001 

47° 

1.8848 

1.3329 

80.4° 

3.1928 

1.0375 

89.7° 

6.6385 

1.0001 

48° 

1.9011 

1.3238 

80.6° 

3.2132 

1.0363 

89.8° 

7.0440 

1.0000 

49° 

1.9180 

1.3147 

80.8° 

3.2340 

1.0350 

89.9° 

7.7371 

1.0000 

32  TABLES 

Common  Logarithms  of  T{n)  for  Values  of  n  between  1  and  2 


t 


n 

log,or(n) 

n 

iogior(«) 

n 

log,or(n) 

n 

iog,or(7i) 

n 

log,or(n) 

1.01 

1.9975 

1.21 

1.9617 

1.41 

T.9478 

1.61 

1.9517 

1.81 

1.9704 

1.02 

1.9951 

1.22 

1.9605 

1.42 

1.9476 

1.62 

T.9523 

1.82 

1.9717 

1.03 

T.9928 

1.23 

1.9594 

1.43 

1.9475 

1.63 

1.9529 

1.83 

1.9730 

1.04 

1.9905 

1.24 

1.9583 

1.44 

1.9473 

1.64 

1.9536 

1.84 

1.9743 

1.05 

1.9883 

1,25 

1.9573 

1.45 

1.9473 

1.65 

1.9543 

1.85 

1.9757 

1.06 

1.9862 

1.26 

1.9564 

1.46 

1.9472 

1.66 

1.9550 

1.86 

1.9771 

1.07 

1.9841 

1.27 

1.9554 

1.47 

1.9473 

1.67 

1.9558 

1.87 

1.9786 

1.08 

1.9821 

1.28 

1.9546 

1.48 

1.9473 

1.68 

1.9566 

1.88 

1.9800 

1.09 

1.9802 

1.29 

1.9538 

1.49 

T.9474 

1.69 

1.9575 

1.89 

1.9815 

1.10 

1.9783 

1.30 

1.9530 

1.50 

1.9475 

1.70 

T.9584 

1.90 

1.9831 

1.11 

1.9765 

1.31 

1.9523 

1.51 

1.9477 

1.71 

1.9593 

1.91 

1.9846 

1.12 

1.9748 

1.32 

1.9516 

1.52 

1.9479 

1.72 

1.9603 

1.92 

1.9862 

1.13 

1.9731 

1.33 

1.9510 

1.53 

1.9482 

1.73 

1.9613 

1.93 

1.9878 

1.14 

1.9715 

1.34 

1.9505 

1.54 

T.9485 

1.74 

1.9623 

1.94 

1.9895 

1.15 

1.9699 

1.35 

1.9500 

1.55 

1.9488 

1.75 

1.9633 

1.95 

1.9912 

1.16 

1.9684 

1.36 

1.9495 

1.56 

1.9492 

1.76 

1.9644 

1.96 

1.9929 

1.17 

1.9669 

1.37 

1.9491 

1.57 

1.9496 

1.77 

1.9656 

1.97 

1.9946 

1.18 

1.9655 

1.38 

1.9487 

1.58 

1.9501 

1.78 

1.9667 

1.98 

T.9964 

1.19 

1.9642 

1.39 

1.9483 

1.59 

1.9506 

1.79 

1.9679 

1.99 

1.9982 

1.20 

1.9629 

1.40 

1.9481 

1.60 

1.9511 

1.80 

1.9691 

2.00 

0.0000 

'4 

rr(5!  +  i)  =  3.r(z),  if  «>o;  r(2)  =  r(i)  =  in 

\  [r  (x) '  r(l  -  X)]  =  TT/sin  jra;,  if  1  >  a^  >  0.       J 


If  the  values  of  an  analytic  function,  f{x),  are  given  in  a  table  for  consecu- 
tive values  of  the  argument,  x,  with  the  constant  interval  d,  and  if  h  =  kd, 
where  k  is  any  desired  fraction, 

/(«  +  .)=/(„)  +  . .A,  +  *-<^.A,  +  *(^:i^ii^.A3  +  .... 

where /(a)  is  any  tabulated  value. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 
LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 
on  the  date  to  which  renewed. 
;     Renewed  books  are  subject  to  immediate  recall. 


:^ 


"irmj^ 


RECEIVED 


0CT2r6D-iPM 


i-OAN  DEPT. 


■tTF^ 


•'■h 


? 


DBRARY  USE 


^J         7kjO: 


JAN  7    lytiZ 


HQV3-i9e6  3  3 


IN  STACKS 


OCT  20 1966 


LD  21A-50m-8/57 
(C8481sl0)476B 


General  Library 

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